On the parabolic Fatou domains

Let f be a rational map with an infinitely-connected fixed parabolic Fatou domain U. We prove that there exists a rational map g with a completely invariant parabolic Fatou domain V, such that (f, U) and (g, V) are conformally conjugate, and each non-singleton Julia component of g is a Jordan curve which bounds a superattracting Fatou domain of g containing at most one postcritical point. Furthermore, we show that if the Julia set of f is a Cantor set, then the parabolic Fatou domain can be perturbed into an attracting one without affecting the topology of the Julia set.